# Chapter 7: Coordinate Geometry

1. If the point A (x , y) is equidistant from B (4 , 2) and C (-2 , 4). Find the relation between x and y.

|AB| = $$\sqrt{(x-5)^{2}+(y-1)^{2}}$$

= $$\sqrt{x^{2}+16-8x+y^{2}+4-4y}$$

|AC| = $$\sqrt{(x+2)^{2}+(y-5)^{2}}$$

= $$\sqrt{x^{2}+4+4x+y^{2}+16-8y}$$

Since, |AB| = |AC|

$$\Rightarrow$$ $$x^{2}+y^{2}-8x-4y+20$$ = $$x^{2}+y^{2}+4x-8y+20$$

$$\Rightarrow$$      8y – 4y = 4x + 8x

$$\Rightarrow$$      y = 3x

2. Find the perimeter of a triangle with vertices (0, 8), (0, 0) and (6, 0)

Let the vertices of the $$\bigtriangleup$$ be P(0, 8), Q(0, 0) and R(6, 0)

∴ PQ = $$\sqrt{(0)^{2}+(-8)^{2}}=\sqrt{64}= 8$$

∴ QR = $$\sqrt{(6)^{2}+(0)^{2}}=\sqrt{36}= 6$$

∴ RP = $$\sqrt{(-6)^{2}+(8)^{2}}=\sqrt{100}= 10$$

∴  Perimeter of $$\bigtriangleup$$ = 8 + 6 + 10 = 24 units

3.Find the coordinates of the perpendicular bisector of the line segment joining the points P (1, 4) and Q (2, 3), cuts the y-axis.

Here, O (0, y), P (1, 4) and Q (2, 3)

AO = BO

$$\Rightarrow$$ $$\sqrt{(0-1)^{2}+(y-4)^{2}}$$ = $$\sqrt{(0-2)^{2}+(y-3)^{2}}$$

$$\Rightarrow$$ $$1+y^{2}+16-8y= 4+y^{2}+9-6y$$

$$\Rightarrow$$   8y – 6y = 17 -13

$$\Rightarrow$$   2y = 4

y = 2

∴The required point is (0, 2)

4. The points which divides the line segment joining the points (5, 8) and (1, 2) in ratio 1:2 internally lies in which quadrant?

Now C$$\left ( \frac{1+10}{3}, \frac{2+16}{3} \right )$$

∴ C$$\left ( \frac{11}{3},6 \right )$$

Since, C$$\left ( \frac{11}{3},6 \right )$$ lies in IV quadrant.

5. If A $$\left ( \frac{x}{2},3 \right )$$ is the mid-point of the line segment joining the points B(4,1) and C(8,7). Find the value of x.

∴$$\frac{4+8}{2}=\frac{a}{2}$$

$$\Rightarrow$$ 24 = 2a

$$\Rightarrow$$ a = 12

6. A line intersects the y-axis at point A and B respectively. If (3, 4) is the mid-point of AB. Find the coordinates of A and B.

Here, $$\frac{x+0}{2}=3$$ $$\Rightarrow x= 6$$

$$\frac{0+y}{2}=4$$ $$\Rightarrow x= 8$$

∴ The coordinates of A and B are (0, 8) and (6, 0).

7. Find the fourth vertex S of a parallelogram PQRS whose three vertices are P (-3, 8), Q (5, 4) and R (6, 2).

Here, $$\frac{-3+6}{2}=\frac{5+x}{2}\Rightarrow x=-1$$

$$\frac{8+2}{2}=\frac{4+x}{2}\Rightarrow x=3$$

∴The fourth vertices S of a parallelogram PQRS is S(-1, 3)

8. Find the coordinates of the point which is equidistant from the three vertices of the

$$\bigtriangleup$$POQ.

Point equidistant from the three vertices of  a right angle triangle is the mid-point of hypotenuse.

$$\Rightarrow \left ( \frac{4x+0}{2},\frac{0+4y}{2} \right )\Rightarrow (2x,2y)$$

9. Find the area of a triangle with vertices (x, y + z), (y, z + x) and (z, x + y).

Area of the required

$$\bigtriangleup = \frac{1}{2}|x(z+x-x-y)+y(x+y-y-z) +(y+z-z-x)|$$

= $$\frac{1}{2}|x(z-y)+y(x-z)+z(y-a)|$$

= $$\frac{1}{2}|0|$$ = 0.

10. Find the area of a triangle with vertices P (2, 0), Q (6, 0) and R (5, 4)

Area of the required $$\bigtriangleup PQR$$

= $$\frac{1}{2}|2(0 – 2)+ 6(4 – 0)+ 5(0 – 0)|$$

= $$\frac{1}{2}|24-4|=10$$ Sq. units.

11. If the points O (0, 0), P (3, 4), Q (x, y) are collinear, then write the relation between x and y.

Since, O (0, 0), P (3, 4), Q (x, y) are collinear.

$$\Rightarrow$$ Area of a triangle formed by these points vanishes

∴ $$\frac{1}{2}|0(4 – y)+3(y-0)+x(0-4)|=0$$

$$\Rightarrow y-2x=0\Rightarrow 2x=y$$

12. Find the coordinates of the point O dividing the line segment joining the point P (6, 5)

and Q(2, 10) in the ratio 2:1.

$$x=\frac{2(2)+1(6)}{5}= \frac{4+6}{5}= 2$$

and $$y=\frac{2(10)+1(5)}{5}= \frac{20+5}{5}= 5$$

∴ O(2, 5)

13. The line joining A(4, 3) and B(-3, 6) meets y-axis at C. At what ratio dose C divides the line segment AB?

Since,R lies on y-axis, let C be (0, y)

Suppose that the required ratio be z:1

∴$$x=\frac{-3z+4}{z+1}\Rightarrow 0= -3z+4\Rightarrow z=\frac{4}{3}$$

∴ The required ratio is 4 : 3.

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1. neel
May 24, 2018

2. Parmesh Gour
May 24, 2018

find the value of y for which the distance between a point p[2,-3]andq[10,y]

3. neeraj
May 24, 2018

do you have exercise wise questions

4. Ajitha
March 9, 2018

variety questions